F-zpd algebras and a multilinear Nullstellensatz

Abstract

Let f=f(x1,…,xm) be a multilinear polynomial over a field F. An F-algebra A is said to be f-zpd (f-zero product determined) if every m-linear functional φ:Am→F which preserves zeros of f is of the form φ(a1,…,am)=τ(f(a1,…,am)) for some linear functional τ on A. We are primarily interested in the question whether the matrix algebra Md(F) is f-zpd. While the answer is negative in general, we provide several families of polynomials for which it is positive. We also consider a related problem on the form of a multilinear polynomial g=g(x1,…,xm) with the property that every zero of f in Md(F)m is a zero of g. Under the assumption that m<2d−3, we show that g and f are linearly dependent.